By 'attempts to get an interview' do you mean job application forms filled in or number of CV's sent on spec?
The latter doesn't really count in my opinion because it is only an attempt to get your CV on file in a very small hope that it will be pulled out when the airline in question is recruiting, by which time they probably would have forgotten about it anyway.
Which leaves the second option, the job application. If you have one of these, it is probably fair to say that the airline is recruiting and so you have a fair chance of getting an interview.
It's about putting your energy in to the right places. There are many airlines that don't have an application form or only give you one if they like your CV in the first place...see above... therefore you need to be in the know. This is where networking comes in very handy. By knowing the right people (or at least being on speaking terms with them) you may be able to find out when recruitment is to take place and jump the gun. Usually by the time you hear or read "what questions will greatbigairline ask me" it is too late, they already have their candidates in place. Many of these will be personal recommendations. It has been said many times before. Network.
I realise I labour the point, but I can answer the other point in your question at the same time. When I decided that I would change career to become an airline pilot I knew no one in the industry. I took the modular route (cost) and immediately after my IR got an instructor rating as jobs were few and far between. I instructed for 18 months and had 4 interviews with airlines during that time. All 4 were a RESULT of knowing somebody. Not necessarily a direct recommendation (one of them definately was, another possibly). The people kind enough to help I met training/instructing.
I know what your thinking, and you're right - 3 huge disappointments. However, if it wasn't for those I would not have had the experience to get the last one and believe me when it happens it is the best feeling in the world.
I think good advice is given above, learn what isn't working and keep adapting. Don't send to one airline something that didn't work for another. There is a chance that the first airline got it wrong but it's more likely to be you!
Finally, I can't reccomend strongly enough that you attend one of Pilot Pete's interview seminars. I'm not sure if he is still running them since he got his command, interviewees will be poorer for it if that's the case. you will find a link to his site on pprune somehere.
Good luck